Answer to Question #462 in Calculus for Kyle

Question

It is known that the sequences {xn} and {yn} have a limit. If we form the sequence {x1; y1; x2; y2; … ; xn; yn; … } would it have a limit?

Answer 1

According to the definition of the convergent sequence: lim(n→∞)xn = a ⇔ ∀ε > 0 ∃N = N(ε) ∀n≥ N:|xn - a|<ε.

Thus, if the limits of {xn} and {yn} are different, it will be impossible to find such a to satisfy the condition. In this case the sequence {x1; y1; x2; y2; … ; xn; yn; … } will not have a limit.